Control system and method for dual active bridge dc/dc converters

ABSTRACT

Control system and method for controlling power flow in dual active bridge converters. The control system comprises a compensator, a feedback unit, and a modulator. An outer loop provides a reference signal. The output of the compensator are the control parameters for the modulator. The feedback unit samples the transformer current once every half switching cycle to thereby determine a power level. The power level is used to determine a modulation technique to apply. The modulation technique is one of phase-shift modulation (PSM), variable duty cycle modulation (VDM), or triangular current mode modulation (TCM).

RELATED APPLICATIONS

This application is a Non-Provisional Patent application which claimsthe benefit of U.S. Provisional Application No. 63/050,386 filed Jul.10, 2020.

TECHNICAL FIELD

The present invention relates to dual active bridge (DAB) DC/DCconverters. More specifically, the present invention relates tocontrolling current and voltage in circuits that include DAB converters.

BACKGROUND

Bidirectional DC/DC converters are used in many applications wherebidirectional power flow is required, such as uninterruptible powersupplies (UPSs), electric vehicles (EVs), vehicle-to-grid (V2G) andgrid-to-vehicle (G2V) systems, and energy storage systems. Inparticular, battery charging applications must be capable of operatingin wide range of load conditions. Because of this, converters capable ofachieving stable and robust operation over the entire load range arerequired.

One of the most prominent bidirectional DC/DC converters is the dualactive bridge (DAB) power converter. Typical control systems for DABconverters are closed-loop feedback systems that include a compensatorand a modulator. The modulator creates switching signals based on fourcontrollable parameters of a current in the DAB converter. Theseparameters are identified from an output of the compensator, whichreceives a current signal from the DAB converter and compensates forerror in the current signal. Four control parameters are typically used:phase-shift (φ), duty cycle of bridge A (D_(A)), duty cycle of bridge B(D_(B)), and switching frequency (f_(s)).

FIG. 1 depicts a conventional control method according to the prior art.As can be seen, the control system is a cascaded control system, whichcomprises an outer loop and an inner loop. The feedback of the outerloop is the output voltage of the converter. The outer loop creates thereference signal (reference current) for the inner loop. The feedbackfor the inner loop is the input DC bus current. Thus, the outer loopregulates the output DC voltage, while the inner loop regulates theinput DC current. Typically, linear control techniques andlook-up-tables (LUT) are used for both the outer loop and the innerloop. These linear techniques are optimized for a fixed operating point(chosen to be quiescent operation). As a result, they show poorperformance when the converter operates over wide input/outputconditions and wide load conditions. They may demonstrate sluggishtransient response, poor tracking, and low stability margins for someoperating conditions.

The most common modulation scheme when operating at high power is thephase-shift-modulation scheme (PSM). In PSM, the duty cycles (D_(A) andD_(B)) and the switching frequency (f_(s)) of the converter areconstant. Thus, the phase shift (φ) is solely responsible forcontrolling the power flow in DAB converters. The main advantages ofusing PSM for controlling DAB converters are the simplicity of controland the ability to provide zero-voltage switching (ZVS) for heavy loads.However, DAB converters show poor performance when PSM is used at low ormedium-level power loads.

Other modulation techniques, including variable duty cycle modulation(VDM) and triangular current mode modulation (TCM) may be used at mediumand low load levels, respectively. However, each of these techniques issimilarly limited to a narrow range of inputs. Additionally, thesemodulation schemes are operated with half cycle symmetry. This mayresult in current spikes during transients. Such current spikes areoften harmful to components of the converters, and can also causetemporary saturation of the transformer component of the converter,thereby reducing the overall power efficiency.

Thus, there is a need for a control system and method that performs wellover a range of input conditions and a range of load conditions.Additionally, there is a need for a control method that preventstemporary saturation of the transformer.

SUMMARY

This document discloses a control system and method for controllingpower flow in dual active bridge converters. The control system consistsof cascaded control with an outer loop and an inner loop. The outer loopcreates an inner loop reference signal. The inner loop includes a newcompensator, a feedback unit and a modulator. A feedback unit samplesthe transformer primary current every half switching cycle. The outputof this feedback unit is either the transformer primary current or theinverse sign of this (depending on the half switching cycle). The innerloop reference signal is subtracted from the output of the feedback unitto create an inner loop error. A geometric-sequence-control (GSC), whichis a model-based predictive current control technique, is used as thecompensator to create the control parameters from this error. Thecontrol parameters are used by the modulator to create the switchinginstants for the switches (i.e. when to switch). The modulationtechnique is one of phase-shift modulation (PSM), variable duty cyclemodulation (VDM), or triangular current mode modulation (TCM). Thecompensator inside the control system allows the inner loop error toapproach zero in a fast and seamless manner.

In a first aspect, this document discloses a control system for a dualactive bridge (DAB) DC/DC converter, said control system comprising: afeedback unit; a compensator; and a modulator, wherein said feedbackunit measures a transformer current from a bridge component of said DABconverter as input and wherein an inner loop reference current issubtracted from an output of said feedback unit; wherein an output ofsaid subtraction is passed to said compensator; wherein said compensatordetermines control parameters for said modulator; and wherein saidmodulator applies a modulation technique based on said controlparameters to change a switching pattern to thereby control saidtransformer current and to thereby control a power flow in said circuit.

In another embodiment, this document discloses a control system whereinsaid modulation technique is one of: phase-shift modulation (PSM),variable duty cycle modulation (VDM), and triangular current modemodulation (TCM).

In another embodiment, this document discloses a control system whereinsaid feedback unit samples said transformer current once every halfswitching cycle.

In another embodiment, this document discloses a control system whereinsaid compensator updates said control parameters once every halfswitching cycle.

In another embodiment, this document discloses a control system whereinsaid modulator updates said switching pattern every half switchingcycle.

In another embodiment, this document discloses a control system whereinsaid transformer current is modified based on an output of saidmodulator.

In another embodiment, this document discloses a control system whereinsaid compensator determines said control parameters using ageometric-sequence-control (GSC) method.

In another embodiment, this document discloses a control system whereinsaid GSC method is based on an error in said inner loop referencecurrent.

In another embodiment, this document discloses a control system whereinsaid GSC method is applied such that an error in said inner loopreference current decreases in a geometric sequence progression oversuccessive half switching cycle samplings, and wherein said controlparameters are by-products of said GSC method.

In another embodiment, this document discloses a control system whereinsaid control parameters comprise at least one of: a phase shift of theconverter, a duty cycle of said bridge component, and a duty cycle ofanother bridge component of said DAB converter.

In another embodiment, this document discloses a control system whereinsaid system conforms to at least one of the following conditions:

-   -   when a power level within said circuit is below a first        threshold, said modulation technique is triangular current mode        modulation (TCM);    -   when said power level is between said first threshold and a        second threshold, said modulation technique is variable duty        cycle modulation (VDM), said second threshold being higher than        said first threshold; and    -   when said power level is above said second threshold, said        modulation technique is phase-shift modulation (PSM).

In a second aspect, this document discloses controlling a power flowwithin a circuit comprising a dual active bridge (DAB) DC/DC converter,said method comprising the steps of: receiving a current from a bridgecomponent of said DAB converter as input for a compensator; measuring acurrent from a transformer component of said DAB converter; based onsaid current, determining control parameters for a modulator; and usingsaid modulator, applying a modulation technique based on said controlparameters, to thereby modify said current and to thereby control saidpower flow in said circuit.

In another embodiment, this document discloses a method wherein saidmodulation technique is one of: phase-shift modulation (PSM), variableduty cycle modulation (VDM), and triangular current mode modulation(TCM).

In another embodiment, this document discloses a method wherein saidcontrol parameters are determined by applying ageometric-sequence-control (GSC) method

In another embodiment, this document discloses a method wherein said GSCmethod is based on inner loop current error.

In another embodiment, this document discloses a method wherein saidcontrol parameters are determined by a compensator.

In another embodiment, this document discloses a method wherein saidcompensator updates said control parameters once every half switchingcycle.

In another embodiment, this document discloses a method wherein said GSCmethod is applied such that an inner loop current error decreases in ageometric sequence progression over successive half switching cyclesamplings, and wherein said control parameters are by-products of saidGSC method.

In another embodiment, this document discloses a method wherein saidcontrol parameters comprise at least one of: a phase shift of theconverter, a duty cycle of said bridge component, and a duty cycle ofanother bridge component of said DAB converter.

In another embodiment, this document discloses a method wherein saidmethod conforms to at least one of the following conditions:

-   -   when a power level within said circuit is below a first        threshold, said modulation technique is triangular current mode        modulation (TCM);    -   when said power level is between said first threshold and a        second threshold, said modulation technique is variable duty        cycle modulation (VDM), said second threshold being higher than        said first threshold; and    -   when said power level is above said second threshold, said        modulation technique is phase-shift modulation (PSM).

In a further aspect, the present invention provides non-transitorycomputer-readable media having encoded thereon computer-readable andcomputer-executable instructions that, when executed, implement a methodfor controlling a power flow within a circuit comprising a dual activebridge (DAB) DC/DC converter, said method comprising the steps of:receiving a current from a bridge component of said DAB converter asinput for a compensator; measuring a current from a transformercomponent of said DAB converter; based on said current, determiningcontrol parameters for a modulator; and using said modulator, applying amodulation technique based on said control parameters, to thereby modifysaid current and to thereby control said power flow in said circuit.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said modulation technique is one of:phase-shift modulation (PSM), variable duty cycle modulation (VDM), andtriangular current mode modulation (TCM).

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said control parameters are determinedby applying a geometric-sequence-control (GSC) method.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said GSC method is based on inner loopcurrent error.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said control parameters are determinedby a compensator.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said compensator updates said controlparameters once every half switching cycle.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said GSC method is applied such that aninner loop current error decreases in a geometric sequence progressionover successive half switching cycle samplings, and wherein said controlparameters are by-products of said GSC method.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein said control parameters comprise atleast one of: a phase shift of the converter, a duty cycle of saidbridge component, and a duty cycle of another bridge component of saidDAB converter.

In another embodiment, this document discloses non-transitorycomputer-readable media wherein the method implemented by the executionof the instructions encoded thereon conforms to at least one of thefollowing conditions: when said power level is below a first threshold,said modulation technique is triangular current mode modulation (TCM);when said power level is between said first threshold and a secondthreshold, said modulation technique is variable duty cycle modulation(VDM), said second threshold being higher than said first threshold; andwhen said power level is above said second threshold, said modulationtechnique is phase-shift modulation (PSM).

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be described by reference to thefollowing figures, in which identical reference numerals refer toidentical elements and in which:

FIG. 1 is a block diagram of a conventional DAB DC/DC converter and aconventional proportional-integral (PI) control system, according to theprior art;

FIG. 2 is a block diagram of a DAB DC/DC converter according to anaspect of the present invention;

FIG. 3A is an expanded block diagram of a conventional DAB converter,according to the prior art;

FIG. 3B is a schematic diagram of the equivalent circuit of a DABconverter, according to the prior art;

FIG. 4 is a plot showing three steady state waveforms (specifically,v_(p), kv_(s), and i_(p)), in conventional DAB converters, according tothe prior art;

FIG. 5A is a 3-dimensional illustration of the zero-voltage switching(ZVS) region of the conventional DAB converter, according to the priorart;

FIG. 5B is a cross-section of FIG. 5A at φ=0.5;

FIG. 5C is a cross-section of FIG. 5A at φ=0.2;

FIGS. 6A and 6B are plots showing waveforms produced by triangularcurrent mode modulation, according to the prior art;

FIG. 7 is a block diagram illustrating an inner loop of the controlsystem according to an aspect of the present invention, when TCM isused;

FIG. 8 is a block diagram illustrating an inner loop of the controlsystem according to an aspect of the present invention, when VDM isused;

FIG. 9 is a block diagram illustrating an inner loop of the controlsystem according to an aspect of the present invention, when PSM isused;

FIG. 10 is a plot showing an exemplary transient response of the DABconverter according to the prior art;

FIGS. 11A and 11B are plots showing i_(p) with different phase-shiftsfor each half cycle;

FIG. 12 is a plot showing a dynamic response of the DAB converteraccording to the present invention;

FIG. 13 is a plot showing steady-state current waveforms associated withcurrent and phase shift;

FIGS. 14A to 14F are plots showing asymmetric waveforms;

FIGS. 15A and 15B are plots showing i_(p) with different duty cycles ofBridge A at each half cycle;

FIGS. 16A and 16B are plots showing i_(p) with different duty cycles ofBridge B at each half cycle;

FIGS. 17A and 17B are plots showing i_(p) with different φ[n] s for eachhalf cycle when v_(a)>kv_(b);

FIGS. 18A and 18B are plots showing i_(p) with different φ[n] s for eachhalf cycle when v_(a)<kv_(b);

FIGS. 19A to 19D depict the transient response of the control system ofFIG. 2;

FIGS. 20A to 20C depict the performance of the control system of FIG. 2when the series inductor is varied by 100%;

FIG. 21 is a plot comparing the transient response of the control systemof FIG. 2 with the response of conventional PI controllers;

FIG. 22 illustrates the transient operation of a DAB converter with aconventional control system, according to the prior art;

FIG. 23 depicts the steady-state operation of a DAB converter with apower flow from port B to port A, according to the prior art; and

FIG. 24 is a flowchart detailing a method according to one aspect of theinvention.

DETAILED DESCRIPTION

For clarity, Table 1 outlines the meaning of terms and symbols usedthroughout this document:

TABLE 1 Acronym/Symbol Meaning k transformer turns ratio L_(s) DAB highfrequency network inductance ƒ_(s) switching frequency T_(s) switchingperiod V_(α) DC voltage of port A V_(b) DC voltage of port B ν_(α, [n])voltage of port A for half cycle [n] ν_(b [n]) DC voltage of port A forhalf cycle [n] ν_(p) transformer primary voltage ν_(s) transformersecondary voltage i_(p) transformer primary current i[n] sample currentof the transformer primary current at the beginning of half cycle [n]Δi[n] current error of half cycle [n] di[n] current difference betweeni[n + 2] and i[n] (i [n + 2] − i[n]) t₀[n] timing instant when v_(p) ischanged to + V_(α) − V_(α) in half cycle [n] t₁[n] time interval betweent₀[n] and when ν_(s) is changed to zero in half cycle [n] t₂[n] timeinterval between the end of interval t₁[n] and when ν_(s), is changedt₃[n] time interval between the end of interval t₂[n] and when ν_(p) ischanged P output power P_(max) maximum possible power level P_(BC) powerlevel that corresponds to φ = φ_(BC) in PSM P_(ZVS,min) lower powerlevel in which ZVS is achieved with VDM P_(Δ,max) maximum achievablepower with TCM D_(α) duty cycle of bridge A D_(α)[n] duty cycle ofbridge A for half cycle [n] D_(b) duty cycle of bridge B D_(bmin) dutycycle of bridge B when lowest possible operating power that ZVS can beachieved with VDM when V_(α) < kV_(b) D_(α),_(min) duty cycle of bridgeA when lowest possible operating power that ZVS can be achieved with VDMwhen V_(α) > kVb D_(b)[n] duty cycle of bridge B for half cycle [n] φphase-shift φ [n] phase-shift for half cycle [n] φ_(BC) minimumphase-shift in which ZVS is achieved with PSM$m_{1} = \frac{V_{a} + {kV_{b}}}{L_{s}}$ slope of current$m_{2} = \frac{{V_{a}1} - {kV_{b}}}{L_{s}}$ slope of current$m_{3} = \frac{- {kV}_{b}}{L_{s}}$ slope of current$m_{4} = \frac{V_{a}}{L}$ slope of current x[n] in general, x[n] is thevalue of x for half cycle [n] PSM phase-shift modulation VDM variableduty cycle modulation TCM triangular current mode modulation ZVSzero-voltage switching

Additionally, to better understand the present invention, the reader isdirected to the listing of citations at the end of this description. Forease of reference, these citations and references have been referred toby their listing number throughout this document. The contents of thecitations in the list at the end of this description are herebyincorporated by reference herein in their entirety.

The present invention provides a control system and method that allowsDAB converters to operate more effectively at a wide range of input andload conditions. In particular, the present invention allows the DABconverter to apply different modulation techniques based on the controlparameters identified. That is, the present invention applies PSM, VDM,or TCM based on parameters of the received signal and feeds back themodulated signal to more effectively compensate for error and controlpower flows. The compensator of the inner loop of the control systemapplies a geometric sequence control (GSC) method to determine thecontrol parameters of the modulator. The operating power of the DABconverter determines which modulation technique to apply. Additionally,the control parameters used by the GSC are updated by sampling the inputsignal every half switching cycle (asymmetric half cycle modulation).This asymmetric half cycle modulation ensures zero DC current in thetransformer winding and helps to prevent transformer saturation.

FIG. 2 is a block diagram showing an embodiment of control system 10 fora DAB converter according to the present invention. It should beunderstood that this diagram represents a prototype circuit that wasused in experimentation, as discussed in greater detail below.Accordingly, certain design choices were made that should not be seen aslimiting the invention. As one example, the configuration of the gatesin the bottom left of this diagram should not be taken to limit theinvention. Any gate configuration that produces a suitably modifiedsignal can be used.

The compensator 30 receives an output current signal (i_(b)(t)) from oneof the bridge components of the DAB converter (in this case, from bridgeB). The output of this compensator 30 is passed to the control unit 40.A primary current is measured at a transformer component of said DABconverter (i_(p)(t)) and modified via a series of gates, to therebyproduce a modified primary current. A reference signal for the GSC loop(i.e., for the inner loop) is created by an outer loop control. Thisouter loop control includes i_(b)[n] as feedback, I_(b,ref) as thereference load, and a compensator to create the reference signal. Thecurrent error created by subtracting the reference signal from themodified transformer (i[n−1] or −i_([)n−1]) is fed to the GSC scheme tocreate the control parameters φ, D_(a), and D_(b). These parameters arepassed to the modulator 50 and are used to determine a modulationtechnique to be applied (either TCM, VDM, or PSM). The signal identifiedas “mode”, which is generated inside the modulator 50 and fed back tothe control unit 40, identifies which modulation technique should beapplied, based on the calculated power level determined from the controlparameters. The outputs of the modulator are the switching signals forthe switches (S_(AN) and S_(BN)) and a sampling signal (ss), which isfed back to the inner loop and used as a basis to determine half cycleswitching instants (i.e., when to switch) for measuring the transformerprimary current.

It should be clear that there are two compensators in the controlsystem: an outer loop compensator/feedback unit 30 and an inner loopcompensator 40. In one aspect of the present invention, the inner loopcompensator uses a geometric-sequence-control (GSC) method while thefeedback unit samples the transformer current once every half cycle andmodifies it. Then, the inner loop reference signal (which is provided asoutput from the outer loop) is subtracted from this modified signal tocreate the inner loop error. The inner loop compensator 40 uses thisinner loop error to determine and change the control parameters. Thecontrol parameters are used by the modulator 50 to create and change theswitching pattern of the switches and to thereby modify the transformercurrent. This thereby changes the power level.

Conventional DAB Converters

FIG. 3A depicts the schematic diagram of a typical DAB DC/DC converter.DAB

DC/DC converters are used as an interface to allow bidirectional powerflow between two DC buses (V_(a) and V_(b)). Due to the four controllingparameters of DAB converters, the shape of the transformer current isvery flexible, in that, by utilizing a combinations of these controllingparameters, ZVS and low RMS current can be achieved. RMS current istypically lower when the current is close to a square wave shape.

DAB converters consist of two active bridges (Bridge A and Bridge B), ahigh frequency network (a transformer with series inductance), and twofilter networks (C_(A) and C_(B)). The two active bridges (A and B)create quasi-square wave voltages (v_(p) and v_(s)) at the two ends ofthe high frequency network. FIG. 3B depicts the equivalent circuit of aDAB converter (in a lossless conversion), where n is the transformerturns ratio, L_(s) is the series inductance, and i_(p) is the current inthe transformer primary winding. Based on the switching pattern, v_(p)can hold three states: +V_(a), −V_(a), and 0; similarly v_(s) has threestates: +V_(b), −V_(b), and 0. The current, i_(p), is calculated as:

$\begin{matrix}{{i_{p}(t)} = {{\frac{1}{L_{s}}{\int_{t_{0}}^{t}{\left( {v_{p} - {kv}_{s}} \right){dt}}}} + {i_{p}\left( t_{0} \right)}}} & (1)\end{matrix}$

In steady-state operation,

$\begin{matrix}{{{v_{p}(t)} = {v_{p}\left( {t + \frac{T_{s}}{2}} \right)}},{and}} & (2) \\{{v_{s}(t)} = {{v_{s}\left( {t + \frac{T_{s}}{2}} \right)}.}} & (3)\end{matrix}$

When there is no DC current present in the transformer winding(i_(p)=0), i_(p)(t) is also equal to

$\begin{matrix}{{i_{p}(t)} = {{i_{p}\left( {t + \frac{T_{s}}{2}} \right)}.}} & (4)\end{matrix}$

FIG. 4 depicts three steady state waveforms (specifically, v_(p),kv_(s), and i_(p)), in conventional DAB converters, when ZVS is achievedfor all the switches (that is, where the DC current of i_(p) is zero).In FIG. 4, t₀ is defined as the instant when S_(A1) is turning on andS_(A2) has turned off, while S_(A3) is off and S_(A4) is on. t₁ isdefined as the instant when S_(B1) is turning on and S_(B2) has turnedoff while S_(B3) and S_(B4) are on and off. The ZVS criteria representedin this figure obey the following relations:

i _(p)(t ₀)<0  (5)

i _(p)(t ₁)>0  (6)

1−(D _(a) +D _(b))<φ<D _(a) +D _(b)  (7)

The steady-state value of i_(p)(t₀), i_(p)(t₁), and the output powerflow P in a lossless conversion can be derived as follows:

$\begin{matrix}{\mspace{76mu}{{{i_{p}\left( t_{0} \right)}\lbrack n\rbrack} = {\frac{1}{2L_{s}f_{s}}\left( {{\left( {- {D_{a}\lbrack n\rbrack}} \right)V_{a}} + {\left( {1 - {D_{a}\lbrack n\rbrack} - {\varphi\lbrack n\rbrack}} \right){kV}_{b}}} \right)}}} & (8) \\{\mspace{76mu}{{i_{p}\left( t_{1} \right)} = {\frac{1}{2L_{s}f_{s}}\left( {{\left( {{D_{b}{p\lbrack n\rbrack}} + \varphi - 1} \right)V_{a}} + {\left( {D_{b}\lbrack n\rbrack} \right){kV}_{b}}} \right)}}} & (9) \\{{P = {\frac{{kV}_{a}V_{b}}{2\mspace{14mu} L_{s}f_{s}}\left( {\frac{1}{4} - \left( {\left( {{D_{a}\lbrack n\rbrack} - \frac{1}{2}} \right)^{2} + \left( {{D_{b}\lbrack n\rbrack} - \frac{1}{2}} \right)^{2} + \left( {{\varphi\lbrack n\rbrack} - \frac{1}{2}} \right)^{2}} \right)} \right)}}\mspace{76mu}{where}} & (10) \\{\mspace{76mu}{{{\varphi\lbrack n\rbrack} = {\frac{1}{2} + {{t_{1}\lbrack n\rbrack}f_{s}} + {{t_{2}\lbrack n\rbrack}f_{s}} - {{t_{3}\lbrack n\rbrack}f_{s}}}},{0 < {\varphi\lbrack n\rbrack} < 0.5}}} & (11) \\{\mspace{76mu}{{D_{a}\lbrack n\rbrack} = {{{t_{3}\lbrack n\rbrack}f_{s}0} < {D_{a}\lbrack n\rbrack} < 0.5}}} & (12) \\{\mspace{76mu}{{D_{b}\lbrack n\rbrack} = {{\frac{1}{2} + {{t_{1}\lbrack n\rbrack}f_{s}} - {{t_{2}\lbrack n\rbrack}f_{s}0}} < {D_{b}\lbrack n\rbrack} < 0.5}}} & (13)\end{matrix}$

where φ[n] is the phase shifts for the n^(th) half cycle, and D_(a)[n]and D_(b)[n] are the duty cycles for the n^(th) cycle of bridge A andbridge B, respectively. Thus, ½φT_(s) is the time difference between thecenter points of the the two quasi-square waveform: v_(p) and v_(s).

Using (8) and (9), the ZVS criteria can be rewritten as:

$\begin{matrix}\left. {{i_{p}\left( t_{0} \right)} < 0}\rightarrow{\frac{V_{a}}{{kV}_{b}} > \frac{1 - \varphi - D_{a}}{D_{a}}} \right. & (14) \\\left. {{i_{p}\left( t_{1} \right)} > 0}\rightarrow{\frac{{kV}_{b}}{V_{a}} > \frac{1 - \varphi - D_{b}}{D_{b}}} \right. & (15) \\{{1 - \left( {D_{a} + D_{b}} \right)} < \varphi < {D_{a} + D_{b}}} & (16)\end{matrix}$

FIG. 5A is a 3-dimensional illustration of the ZVS region of theconventional DAB converter, when V_(a)=90 V_(DC), V_(b)=110 V_(DC),f_(s)=250 kHz, n=0.9, and L_(s)=10 μH. This figure depicts all thepossible combinations of the control parameters D_(a), D_(b), φ thatcomply with the constraints given in (14), (15), and (16). The color baron the right shows the power levels of the converter for each contour.FIG. 5B is a 2-dimensional representation that shows the cross sectionof FIG. 5A for φ=0.5. Similarly, FIG. 5C shows the cross section of FIG.5A for φ=0.2. The bar on the right of each figure shows the power levelfor each contour.

Hybrid Modulation Scheme Phase-Shift Modulation (PSM)

In PSM, the duty cycles D_(a) and D_(b) are equal to 0.5. That is, thephase-shift, φ, is solely responsible for controlling power flow. TheZVS criteria for PSM can be derived as:

$\begin{matrix}{0 < \varphi < 0.5} & (17) \\{{\min\left\{ {\frac{V_{a}}{{kV}_{b}},\frac{{kV}_{b}}{V_{a}}} \right\}} > {1 - {2\varphi}}} & (18)\end{matrix}$

The extreme cases, where φ=0.5 and φ=0 correspond to the maximum andminimum power flow with the PSM modulation, respectively, can be writtenas follows:

$\begin{matrix}{{{\min\left\{ {\frac{V_{a}}{{kV}_{b}},\frac{{kV}_{b}}{V_{a}}} \right\}} > {0\varphi}} = 0.5} & (19) \\{{{\min\left\{ {\frac{V_{a}}{{kV}_{b}},\frac{{kV}_{b}}{V_{a}}} \right\}} > {1\varphi}} = 0} & (20)\end{matrix}$

The minimum phase-shift for which ZVS is achieved with PSM is called the“boundary condition phase-shift” (φ_(BC)). ZVS is lost for phase-shiftslower than φ_(BC). Equation (18) can be rewritten to find φ_(BC) asfollows:

$\begin{matrix}{\varphi_{BC} = {\frac{1}{2}\left( {1 - {\min\left\{ {\frac{V_{a}}{{kV}_{b}},\frac{{kV}_{b}}{V_{a}}} \right\}}} \right)}} & (21)\end{matrix}$

Thus, in the hybrid modulation scheme of the present invention, PSM canbe effectively used for phase-shifts within the [φ_(BC) to 0.5]interval. That is, PSM is used for operating conditions where the powerflow ranges between [P_(BC) to P_(max)], where P_(BC) corresponds to thepower flow when φ=φ_(BC) and P_(max) corresponds to the power flow whenφ=0.5. For power levels lower than P_(BC), which correspond tophase-shifts lower than φ_(BC), VDM is applied.

Variable Duty Cycle Modulation (VDM)

A form of VDM is used that can achieve ZVS at power levels lower thanP_(BC) by only using one control parameter. From (10), it can beinferred that, in order to decrease the power flow, either φ, D_(a), orD_(b) has to be decreased. In the VDM of the present invention, φ iskept constant at φ=φ_(BC) and one of the duty cycles (D_(a) or D_(b))decreases while the other remains at 0.5. Decreasing both duty cycles atthe same time while φ=φ_(BC) would cause the loss of ZVS. The relationbetween kV_(b) and V_(a) determines which of the duty cycles (D_(a) orD_(b)) can be changed to maintain ZVS, as follows:

-   -   Case 1. V_(a)<kV_(b): In this case, D_(a)=0.5 and D_(b) is used        to control the power flow.    -   Case 2. kV_(b)<V_(a): In this case, D_(b)=0.5 and D_(a) is used        to control the power flow.    -   Case 3. kV_(b)=V_(a): In this case, PSM covers the whole range        of power levels since φ_(BC)=0 and P_(BC)=0. Therefore, there is        no need for VDM in this scenario.

Case 1. V_(a)<kV_(b)

According to (21), φ_(BC) can be calculated as:

$\begin{matrix}{\varphi_{BC} = {\frac{1}{2}\left( {1 - \frac{V_{a}}{{kV}_{b}}} \right)}} & (22)\end{matrix}$

Phase-shifts lower than φ_(BC) violate the ZVS criteria introduced in(14). That is, if φ<φ_(BC), i_(p)(t₀) will become positive. Similarly,reducing D_(a) while φ=φ_(BC) violates the ZVS criteria in (14). Thus,in this case, D_(b) is the only control parameter while φ and D_(a)remain at φ_(BC) and 0.5, respectively. As D_(b) decreases, the ZVScriteria in (15) and (16) becomes more stringent. The lowest power level(P_(ZVS,min)) under ZVS is when φ=φ_(BC) and D_(b)=D_(bmin), where

D _(bmin)=0.5−φ_(BC)  (23)

and P_(ZVS,min) can be calculated from:

$\begin{matrix}{P_{{ZVS},\min} = {\frac{{kV}_{a}V_{b}}{2\mspace{14mu} L_{s}f_{s}}\left( {\varphi_{BC} - {2\varphi_{BC}^{2}}} \right)}} & (24)\end{matrix}$

Case 2. kV_(b)<V_(a)

According to (21), φ_(BC) can be calculated as:

$\begin{matrix}{\varphi_{BC} = {\frac{1}{2}\left( {1 - \frac{{kV}_{b}}{V_{a}}} \right)}} & (25)\end{matrix}$

In this case, D_(a) is required to control the power flow. Similar tothe previous case, D_(amin) can be calculated as

D _(amin)=0.5−φ_(BC)  (26)

and P_(ZVS,min) can be calculated as in (24).

Thus, with the hybrid modulation, ZVS can be achieved over(P_(ZVS,min)<P<P_(max)), where VDM is used for P_(ZVS,min)<P<P_(BC) andPSM is used for P_(BC)<P<P_(max).

Case 3. kV_(b)=V_(a)

Again, as PSM covers the whole range of power levels in this case, thereis no need for VDM in this scenario.

Triangular Current Mode Modulation (TCM)

At low power operations, TCM modulation shows the lowest RMS current inthe transformer primary winding. Thus, this modulation scheme isimplemented for power levels below P_(ZVS,min)(P<_(ZVS,min)). The keywaveforms of the TCM modulation are depicted in FIGS. 6A and 6B. Atsteady-state operation, the control variables for all half cycles areequal

${\left( {{{\varphi\lbrack n\rbrack} = {\varphi\left\lbrack {n + 1} \right\rbrack}},{{D_{a}\lbrack n\rbrack} = {D_{a}\left\lbrack {n + 2} \right\rbrack}},{{{and}\mspace{14mu}{D_{b}\lbrack n\rbrack}} = {D_{b}\left\lbrack {n + 1} \right\rbrack}}} \right)\mspace{14mu}{and}\mspace{14mu}{i_{p}(t)}} = {{i_{p}\left( {t + \frac{T_{s}}{2}} \right)}.}$

The steady-state equations of power flow and the control variables inthe TCM scheme are given in Ref. [1], noted below. These equations canbe expressed as:

$\begin{matrix}{\begin{matrix}{P = \frac{\varphi^{2}{V_{a}\left( {kV_{b}} \right)}^{2}}{L_{s}{f_{s}\left( {V_{a} - {k\; V_{b}}} \right)}}} \\{D_{a} = {\frac{- m_{3}}{m_{2}}\varphi}} \\{D_{b} = {D_{a} + \varphi}}\end{matrix}{for}\mspace{14mu}\left\{ \begin{matrix}{P < P_{\Delta,\max}} \\{V_{a} > {kv}_{b}}\end{matrix} \right.} & (27) \\{\begin{matrix}{P = \frac{\varphi^{2}k\;{V_{b}\left( V_{a} \right)}^{2}}{L_{s}{f_{s}\left( {{k\; V_{b}} - V_{a}} \right)}}} \\{D_{a} = {\frac{- m_{4}}{m_{2}}\varphi}} \\{D_{b} = {D_{a} + \varphi}}\end{matrix}{for}\mspace{14mu}\left\{ {\begin{matrix}{P < P_{\Delta,\max}} \\{V_{a} < {kv}_{b}}\end{matrix}\mspace{14mu}{where}} \right.} & (28) \\{m_{2} = \frac{V_{a} - {kV_{b}}}{L_{s}}} & (29) \\{m_{3} = \frac{{- k}V_{b}}{L_{s}}} & (30) \\{m_{4} = \frac{V_{a}}{L_{s}}} & (31)\end{matrix}$

P_(Δ,max) is equal to P_(ZVS,min) introduced in (24)(P_(Δ),max=P_(ZVS,min)); hence, the minimum operating power of the VDMscheme coincides with the maximum power level of the TCM scheme.According to (27) and (28), φ is used to control the power flow and theduty cycles (D_(a) and D_(b)) are calculated as functions of φ. Thus, inone aspect of the present invention, the control is designed such thatφ[n] is determined by the GSC control law and the duty cycles for halfcycle [n] are derived from:

$\begin{matrix}{{D_{a}\lbrack n\rbrack} = {{\frac{k{v_{b}\lbrack n\rbrack}}{{v_{a}\lbrack n\rbrack} - {k{v_{b}\lbrack n\rbrack}}}}{\varphi\lbrack n\rbrack}}} & (32) \\{{D_{b}\lbrack n\rbrack} = {{\frac{v_{a}\lbrack n\rbrack}{{v_{a}\lbrack n\rbrack} - {k{v_{b}\lbrack n\rbrack}}}}{\varphi\lbrack n\rbrack}}} & (33)\end{matrix}$

where v_(a)[n] and v_(b)[n] are the voltages of ports A and B for then^(th) half cycle, respectively.

Geometric Sequence Control (GSC) Method

The GSC method applied by the control unit 40 and the modulator 50 iscapable of accurately and seamlessly tracking the reference signal suchthat the inner current loop error (Δi_(p)[n]) decreases in a geometricsequence progression (hence the name geometric-sequence-control (GSC)).GSC is a recursive nonlinear difference equation, which creates thecontrol parameters. The control variables are updated every halfswitching cycle at the sampling instant. The sampling instants varydepending on the converter operating condition. Table 2 gives a list ofthese instants for all the conditions with respect to the modulationtype and the relationship between v_(a)[n] and kv_(b)[n].

TABLE 2 modulation condition sample instant *PSM v_(a)[n] > kv_(b)[n]v_(p) changes v_(a)[n] < kv_(b)[n] polarity *VDM v_(a)[n] > kv_(b)[n]v_(p) changes from zero to ±v_(a)[n] v_(a)[n] < kv_(b)[n] v_(s) changesfrom zero to ±v_(b)[n] *TCM v_(a)[n] > kv_(b)[n] v_(p) changes from±v_(a)[n] to zero va[n] < kv_(b)[n] v_(s) changes from ±v_(b)[n] to zero

FIG. 7 depicts the inner loop of the control system of one aspect of thepresent invention when TCM is used (for power levels of (P<P_(ZVS,min)).When kV_(b)<V_(a), the GSC control law for the TCM mode is derived asfollows:

$\begin{matrix}{{\varphi\lbrack n\rbrack} = {{\frac{0.5L_{s}f_{s}}{2k{v_{b}\lbrack n\rbrack}}\Delta{i\left\lbrack {n - 1} \right\rbrack}} + {\varphi\left\lbrack {n - 1} \right\rbrack}}} & (34) \\{{D_{a}\lbrack n\rbrack} = {\frac{k{v_{b}\lbrack n\rbrack}}{{v_{a}\lbrack n\rbrack} - {k{v_{b}\lbrack n\rbrack}}}{\varphi\lbrack n\rbrack}}} & (35) \\{{D_{b}\lbrack n\rbrack} = {\frac{v_{a}\lbrack n\rbrack}{{v_{a}\lbrack n\rbrack} - {k{v_{b}\lbrack n\rbrack}}}{\varphi\lbrack n\rbrack}}} & (36)\end{matrix}$

where φ[n−1] is the phase shift of the n−1^(th) half cycle. WhenkV_(b)>V_(a), the GSC control law for the TCM mode is achieved as:

$\begin{matrix}{{\varphi\lbrack n\rbrack} = {{\frac{{0.5}L_{s}f_{s}}{2{v_{a}\lbrack n\rbrack}}\Delta{i\left\lbrack {n - 1} \right\rbrack}} + {\varphi\left\lbrack {n - 1} \right\rbrack}}} & (37) \\{{D_{a}\lbrack n\rbrack} = {\frac{k{v_{b}\lbrack n\rbrack}}{{k{v_{b}\lbrack n\rbrack}} - {v_{a}\lbrack n\rbrack}}{\varphi\lbrack n\rbrack}}} & (38) \\{{D_{b}\lbrack n\rbrack} = {\frac{v_{a}\lbrack n\rbrack}{{k{v_{b}\lbrack n\rbrack}} - {v_{a}\lbrack n\rbrack}}{\varphi\lbrack n\rbrack}}} & (39)\end{matrix}$

FIG. 8 depicts the inner loop of the control system of the presentinvention when VDM is used (for power levels of (P_(ZVS,min)<P<P_(BC)).When kV_(b)<V_(a), the GSC control law for the VDM mode is as follows:

$\begin{matrix}{{D_{a}\lbrack n\rbrack} = {{\frac{0.5L_{s}f_{S}}{{v_{a}\lbrack n\rbrack} + {k{v_{b}\lbrack n\rbrack}}}\Delta{i\left\lbrack {n - 1} \right\rbrack}} + {D_{a}\left\lbrack {n - 1} \right\rbrack}}} & (40) \\{{D_{b}\lbrack n\rbrack} = 0.5} & (41)\end{matrix}$

where D_(a)[n] and D_(a)[n−1] are the duty cycles for n^(th) and(n−1)^(th) half switching cycles of bridge A, respectively. v_(a)[n] isthe voltage of port A for the n^(th) half cycle. When kV_(b)>V_(a), theGSC control law for the VDM mode is as follows:

$\begin{matrix}{{D_{b}\lbrack n\rbrack} = {{\frac{0.5L_{s}f_{s}}{{v_{a}\lbrack n\rbrack} + {k{v_{b}\lbrack n\rbrack}}}\Delta{i\left\lbrack {n - 1} \right\rbrack}} + {D_{b}\left\lbrack {n - 1} \right\rbrack}}} & (42) \\{{D_{a}\lbrack n\rbrack} = 0.5} & (43)\end{matrix}$

where D_(b)[k] and D_(b)[k−1] are the duty cycles for k^(th) and(k−1)^(th) half switching cycles of bridge B, respectively. The hybridmodulator uses these duty cycles and φ=φ_(BC) to create the switchinginstants.

FIG. 9 depicts the inner loop of the control system of one aspect of thepresent invention when PSM is used. For this aspect of the presentinvention, the GSC control law for PSM is defined as follows:

$\begin{matrix}{{\varphi\lbrack n\rbrack} = {{{0.5}\frac{L_{s}f_{s}}{k{v_{b}\lbrack n\rbrack}}\Delta{i\left\lbrack {n - 1} \right\rbrack}} + {\varphi\left\lbrack {n - 1} \right\rbrack}}} & (44)\end{matrix}$

where φ[n] and φ[n−1] are the phase-shifts for n^(th) and (n−1)^(th)half switching cycles, respectively. Δi_(p,t) ₀ [k−1] is the discretecurrent error (i.e. Δi_(p)=i_(ref)−i_(p)) for the (n−1)^(th) half cycleat t=t₀, and v_(b)[n] is the voltage of port B in the n^(th) half cycle.

Derivation of GSC for PSM

FIG. 10 depicts an exemplary transient response of the DAB converterwhen the phase-shift changes in response to a step change in thereference current (waveforms i_(p), v_(p), and v_(s)), using aconventional control method. As can be seen, the change in thephase-shift (φ[n]) in response to transients creates imbalance in thetransformer current waveform. This imbalance may result in a DCcomponent that saturates the transformer during transients.

Specifically, in FIG. 10, the converter starts the transient from halfcycle [0]. That is, the converter is at steady-state for the half cycles[−2] and [−1]. In the steady-state cycle, φ[−2]=φ[−1] and the magnitudesof the sampled currents (illustrated by solid dots) are equal (i.e.,i[−2]=i[−1]=i[0]). The step change in the reference signal applied atthe switching cycle 1 causes the phase-shift to increase. The increasein the phase-shift φ (i.e., φ[0]=φ[1] & φ[2]=φ[3] & . . . ) results inthe increase in the sampled currents at the beginning of odd cycles(i[1]<i[3]), while the sampled currents at the beginning of even halfcycles do not change (i[0]=i[2]=i[4]). The trajectories of these sampledcurrents are illustrated in FIG. 10. Such asymmetrical growth of i_(p)causes the imbalance between its positive half cycle and its negativehalf cycle, leading to a temporary saturation of the transformer. Themain issue stems from the fact that the phase-shift is constant in everyswitching cycle (two consecutive half cycles) during transient (e.g.,φ[0]=φ[1] & φ[2]=φ[3]). In order to eliminate this issue, the GSC methodfor this aspect of the present invention changes φ every half cycle(i.e., φ[0]≠[1] & φ[2]≠φ[3]).

FIGS. 11A and 11B depict the deviation of i_(p) when there is a changein the phase-shift within a switching cycle (i.e., differentphase-shifts for each half cycle). FIG. 11A depicts the first half cycle[n] when v_(p)>0. To better describe the changes in i_(p) changes whenthere are changes in the phase-shift at each half cycle, φ*[n] isintroduced. φ*[n] can be defined as the only value of phase-shift forhalf cycle [n] that corresponds to a current waveform, where themagnitude of the sampled current at the end of the half cycle [n] isequal to its beginning sampled current (i[n]). This current waveformcorresponding to φ*[n] is depicted by a dashed line, which shows asteady-state current waveform when the reference value is equal to themagnitude of the sampled current at the beginning of the half cycle [n].The solid line in half cycle [n] is the actual current of i_(p), whichcorresponds to a phase-shift of φ[n]. The difference between actualphase-shift and the steady-state phase-shift for half cycle [n] is:

Δφ[n]=φ[n]−φ*[n]  (45)

By using the geometry in FIG. 11, the following can be derived forv_(p)>0:

$\begin{matrix}{\frac{{i\left\lbrack {n + 1} \right\rbrack} + {i\lbrack n\rbrack}}{T_{s}\text{/}2} = {\left( {m_{1} - m_{2}} \right){{\Delta\varphi}\lbrack n\rbrack}}} & (46)\end{matrix}$

where m₁ and m₂ are the slopes and can be calculated as:

$\begin{matrix}{m_{1} = \frac{V_{a} + {kV_{b}}}{L_{s}}} & (47)\end{matrix}$

FIG. 11B depicts the deviation of i_(p) for the half cycle [n] whenv_(p)<0. Similar to the first half cycle, φ*[n] for the second halfcycle is defined such that the current value at the end of the secondhalf cycle is equal to −i[n]. In this case, the following relationshipis derived according to FIG. 11B (for v_(p)<0):

$\begin{matrix}{\frac{{- {i\left\lbrack {n + 1} \right\rbrack}} - {i\lbrack n\rbrack}}{T_{s}\text{/}2} = {\left( {m_{1} - m_{2}} \right){{\Delta\varphi}\lbrack n\rbrack}\mspace{14mu}{where}}} & (48) \\{{\Delta\;{\varphi\left\lbrack {n + 1} \right\rbrack}} = {{\varphi\left\lbrack {n + 1} \right\rbrack} - {\varphi^{*}\left\lbrack {n + 1} \right\rbrack}}} & (49)\end{matrix}$

By using (48) (for the half cycle [n+1]) and (46), the followingrelationship is derived:

$\begin{matrix}{\frac{{- {i\left\lbrack {n + 2} \right\rbrack}} + {i\lbrack n\rbrack}}{T_{s}\text{/}2} = {\left( {m_{1} - m_{2}} \right)\left( {{{\Delta\varphi}\lbrack n\rbrack} + {{\Delta\varphi}\left\lbrack {n + 1} \right\rbrack}} \right)}} & (50)\end{matrix}$

Eq'n. 50 is valid when i[n+2]<i[n+1] or i[n]<i[n+1]. The samerelationship for i[n+2]>i[n+1] or i[n]>i[n+1] can be derived by using(48) for the half cycle [n−1] (n=n−1) and (46), as follows:

$\begin{matrix}{\frac{{i\left\lbrack {n + 2} \right\rbrack} - {i\lbrack n\rbrack}}{T_{s}\text{/}2} = {\left( {m_{1} - m_{2}} \right)\left( {{{\Delta\varphi}\lbrack n\rbrack} + {{\Delta\varphi}\left\lbrack {n + 1} \right\rbrack}} \right)}} & (51)\end{matrix}$

The rate of change between the current samples is defined based on thedifference between the current samples at the beginning of the firsthalf cycle and the end of the consecutive half cycle (i.e. i[n+2] andi[n]) as:

$\begin{matrix}{\frac{d{i\lbrack n\rbrack}}{dt} = \frac{{i\left\lbrack {n + 2} \right\rbrack} - {i\lbrack n\rbrack}}{T_{s}}} & (52)\end{matrix}$

By using (47), (29), and (52) in (50) and (51), the rate of change forthe current samples is derived as:

$\begin{matrix}{\frac{d{i\lbrack n\rbrack}}{dt} = \left\{ \begin{matrix}{\frac{{- k}V_{b}}{L_{s}}\left( {{{\Delta\varphi}\lbrack n\rbrack} + {{\Delta\varphi}\left\lbrack {n + 1} \right\rbrack}} \right)} & {{\begin{matrix}\  \\\ \end{matrix}{i\lbrack n\rbrack}} < {i\left\lbrack {n + 1} \right\rbrack}} \\{\frac{kV_{b}}{L_{s}}\left( {{{\Delta\varphi}\lbrack n\rbrack} + {{\Delta\varphi}\left\lbrack {n + 1} \right\rbrack}} \right)} & {{i\lbrack n\rbrack} > {i\left\lbrack {n + 1} \right\rbrack}}\end{matrix} \right.} & (53)\end{matrix}$

Eq'n. (53) shows the impact of φ on the value of current samples for theconsecutive half cycles [n] and [n+1].

The current error (Δi[n]) is defined as:

$\begin{matrix}{{\Delta{i\lbrack n\rbrack}} = \left\{ \begin{matrix}{i_{ref} + {i\lbrack n\rbrack}} & {\ {{i\lbrack n\rbrack} < {i\left\lbrack {n + 1} \right\rbrack}}} \\{i_{ref} - {i\lbrack n\rbrack}} & {{i\lbrack n\rbrack} > {i\left\lbrack {n + 1} \right\rbrack}}\end{matrix} \right.} & (54)\end{matrix}$

where i_(ref) is the current reference. Thus, the rate of change of thecurrent error is given by:

$\begin{matrix}{\frac{d\Delta{i\lbrack n\rbrack}}{dt} = {\frac{{\Delta{i\left\lbrack {n + 2} \right\rbrack}} - {\Delta{i\lbrack n\rbrack}}}{T_{s}} = \left\{ \begin{matrix}\frac{{+ d}{i\lbrack n\rbrack}}{dt} & {\ {{i\lbrack n\rbrack} < {i\left\lbrack {n + 1} \right\rbrack}}} \\\frac{{- d}{i\lbrack n\rbrack}}{dt} & {\ {{i\lbrack n\rbrack} > {i\left\lbrack {n + 1} \right\rbrack}}}\end{matrix} \right.}} & (55)\end{matrix}$

Eq'n. 55 can be rewritten as:

$\begin{matrix}{\frac{d{i\lbrack n\rbrack}}{dt} = \left\{ \begin{matrix}\frac{d\Delta{i\lbrack n\rbrack}}{dt} & {{i\lbrack n\rbrack} < {i\left\lbrack {n + 1} \right\rbrack}} \\\frac{{- d}\Delta{i\lbrack n\rbrack}}{dt} & {{i\lbrack n\rbrack} > {i\left\lbrack {n + 1} \right\rbrack}}\end{matrix} \right.} & (56)\end{matrix}$

By substituting (56) into (53), the following difference equation isderived for every two consecutive half cycles:

$\begin{matrix}{\frac{d\Delta{i\lbrack n\rbrack}}{dt} = {\frac{{- k}V_{b}}{L_{s}}\left( {{{\Delta\varphi}\lbrack n\rbrack} + {{\Delta\varphi}\left\lbrack {n + 1} \right\rbrack}} \right)}} & (57)\end{matrix}$

The control system is designed such that

$\frac{d\Delta{i\lbrack n\rbrack}}{dt}$

(the error) approaches zero by controlling φ[n].

FIG. 12 depicts the exemplary dynamic response of the DAB converter whenthe GSC scheme of the present invention is used to control φ. Accordingto this figure, the converter is initially operating at steady-state

$\left( {\frac{d\Delta{i\lbrack n\rbrack}}{dt} = 0} \right).$

Then, at the beginning of the half cycle [0], there is a step change ini_(ref) (the new value of i_(ref) is illustrated by an asterisk sign (*)in the figure). The responses of the GSC scheme when the current valueat the beginning of the half cycle is positive are shown at the top ofFIG. 12. The responses when the current value at the beginning of thehalf cycle is negative are shown at the bottom of FIG. 12. As can beseen, the GSC method for this aspect of the present invention creates ahalf cycle delay in response to the step change. That is,φ[0]=φ[−1]=φ[−2] and i[−2]=i[−1]=i[0]=i[1]. In the next half cycle (halfcycle [1]), the GSC changes φ[1] according to its specific control law,and in turn |i[1]|<|i[2]|. In the half cycle [2], the GSC changes φ[2]such that |i[3]|=|i[2]|. This procedure is continued until |i[n]| isrendered zero. That is, in odd half cycles (half cycle [n=2m+1], where mis a natural number), |i[2m+1]|<|i[2m+2]| and in even half cycles (halfcycle [n=2m], where m is a natural number),

$\begin{matrix}\left\{ \begin{matrix}{{{i\left\lbrack {{2m} + 1} \right\rbrack}} = {{i\left\lbrack {2m} \right\rbrack}}} \\{{\Delta{i\left\lbrack {{2m} + 1} \right\rbrack}} = {\Delta{i\left\lbrack {2m} \right\rbrack}}}\end{matrix} \right. & (58)\end{matrix}$

Thus, the magnitude of the sampled current at the end of the even halfcycles [n=2m] is equal to its beginning, which concludes:

$\begin{matrix}\left\{ \begin{matrix}{{\varphi\left\lbrack {2m} \right\rbrack} = {{\varphi^{*}\left\lbrack {2m} \right\rbrack} = {\varphi^{*}\left\lbrack {{2m} + 1} \right\rbrack}}} \\{{{\Delta\varphi}\left\lbrack {2m} \right\rbrack} = 0}\end{matrix} \right. & (59)\end{matrix}$

The dashed line in FIG. 12 shows the trajectory of the odd and evensampled currents in their transition towards the magnitude of i_(ref)and steady-state operation. In the GSC approach, error reduces by halfin every switching cycle. Hence:

Δi[n+2]=0.5Δi[n]  (60)

Eq'n (60) results in a fast transient response. A faster response ispossible by simply changing the 0.5 factor in Eq'n 60 to any valuebetween 0.5 and 1. However, note that changing this factor maycompromise seamless transitions in the dynamic operation. Accordingly,the factor of 0.5 may be preferable, as achieving a balance between fastresponse and smooth transitions.

Accordingly, for the present invention, the GSC control law for PSM(given by (44)) can be derived based on Eq'ns. (58), (59) and (60). Byusing (60) and (55),

$\frac{d\Delta{i\lbrack n\rbrack}}{dt}$

is given by:

$\begin{matrix}{\frac{d\Delta{i\lbrack n\rbrack}}{dt} = {\frac{{\Delta{i\left\lbrack {n + 2} \right\rbrack}} - {\Delta{i\lbrack n\rbrack}}}{T_{s}} = {{- {0.5}}\frac{\Delta{i\lbrack n\rbrack}}{T_{s}}}}} & (61)\end{matrix}$

Then, by inserting (61) into (57), the following relationship is derivedfor each half cycle:

$\begin{matrix}{{{0.5}\frac{\Delta{i\lbrack n\rbrack}}{T_{s}}} = {\frac{kV_{b}}{L_{s}}\left( {{{\Delta\varphi}\lbrack n\rbrack} + {{\Delta\varphi}\left\lbrack {n + 1} \right\rbrack}} \right)}} & (62)\end{matrix}$

Eq'n. (62) for even half cycles results in:

$\begin{matrix}{{{0.5}\frac{\Delta{i\left\lbrack {2m} \right\rbrack}}{T_{s}}} = {\frac{kV_{b}}{L_{s}}\left( {{{\Delta\varphi}\left\lbrack {2m} \right\rbrack} + {{\Delta\varphi}\left\lbrack {{2m} + 1} \right\rbrack}} \right)}} & (63)\end{matrix}$

By rearranging (63), Δφ for odd half cycles is achieved as (from (59),Δφ[2m]=0):

$\begin{matrix}{{{\Delta\varphi}\left\lbrack {{2m} + 1} \right\rbrack} = {{0.5}\frac{f_{s}L_{s}}{kV_{b}}\Delta{i\left\lbrack {2m} \right\rbrack}}} & (64)\end{matrix}$

Eq. (62) results in the same relationship as (64) for odd half cycles.By using (45) when n=2m+1 in (64), the following relationship isderived:

$\begin{matrix}{{{\varphi\left\lbrack {{2m} + 1} \right\rbrack} - {\varphi^{*}\left\lbrack {{2m} + 1} \right\rbrack}} = {{0.5}\frac{f_{s}L_{s}}{kV_{b}}\Delta{i\left\lbrack {2m} \right\rbrack}}} & (65)\end{matrix}$

The control law is derived for odd half cycles by using (59) in (65), asfollows:

$\begin{matrix}{{\varphi\left\lbrack {{2m} + 1} \right\rbrack} = {{{0.5}\frac{f_{s}L_{s}}{kV_{b}}\Delta{i\left\lbrack {2m} \right\rbrack}} + {\varphi\left\lbrack {2m} \right\rbrack}}} & (66)\end{matrix}$

The control law for even half cycles is derived similarly. To this end,a relation between φ*[n], φ*[n+1], and Δφ[n] is beneficial. FIG. 13depicts the steady-state current waveforms associated with φ*[n] andφ*[n+1]. The magnitude of all the sampled currents that correspond toφ*[n] is equal to and the magnitude of all the sampled currents thatcorrespond to φ*[n+1] is equal to |i[n+1]|. Since these waveforms are atsteady-state, (8) can be used to find |i[n]| and [|i[n+1]|]:

$\begin{matrix}{\frac{{i\lbrack n\rbrack}}{T_{s}\text{/}2} = {\frac{1}{L_{s}}\left( {{\left( {+ {0.5}} \right)V_{a}} + {\left( {{\varphi^{*}\lbrack n\rbrack} - {0.5}} \right)kV_{b}}} \right)}} & (67) \\{\frac{{i\left\lbrack {n + 1} \right\rbrack}}{T_{s}\text{/}2} = {\frac{1}{L_{s}}\left( {{\left( {+ {0.5}} \right)V_{a}} + {\left( {{\varphi^{*}\left\lbrack {n + 1} \right\rbrack} - {0.5}} \right)kV_{b}}} \right)}} & (68)\end{matrix}$

Then, by subtracting (67) from (68),

$\begin{matrix}{\frac{{{i\left\lbrack {n + 1} \right\rbrack}} - {{i\lbrack n\rbrack}}}{T_{s}\text{/}2} = {\frac{kV_{b}}{L_{s}}\left( {{\varphi^{*}\left\lbrack {n + 1} \right\rbrack} - {\varphi^{*}\lbrack n\rbrack}} \right)}} & (69)\end{matrix}$

An important property of DAB converters operating in PSM can be derivedby combining (46), (48), and (69) as follows:

2Δφ[n]=φ*[n+1]−φ*[n]  (70)

That is, the error in a conventional DAB converter, without a GSCcontrol system implemented, grows according to Eq'n. (70).

Eq. (70) for n=2m+1 yields:

φ*[2m+1]+Δφ[2m+1]=φ*[2(m+1)]−Δφ[2m+1]  (71)

Combining Eq'n. (45) for n=2m+1, (64), (58), (59), and (71), thefollowing GSC control law for PSM can be derived for even half cycles asfollows:

$\begin{matrix}{{\varphi\left\lbrack {2m^{\prime}} \right\rbrack} = {{\varphi\left\lbrack {{2m^{\prime}} - 1} \right\rbrack} + {{0.5}\frac{f_{s}L_{s}}{kV_{b}}\Delta\;{i\left\lbrack {m^{\prime} - 1} \right\rbrack}}}} & (72)\end{matrix}$

where m′=m+1. Thus, using (66) and (72), the general form of the GSCcontrol law for PSM is derived as:

$\begin{matrix}{{\varphi\lbrack n\rbrack} = {{{0.5}\frac{L_{s}f_{s}}{kV_{b}}\Delta{i\left\lbrack {n - 1} \right\rbrack}} + {\varphi\left\lbrack {n - 1} \right\rbrack}}} & (73)\end{matrix}$

The performance of the GSC control is evaluated when there is DC currentin the transformer winding during transients. (The DC current is definedas the average of the current for a switching cycle.) The DC current isthe result of asymmetry in the current waveform. The asymmetry can occurin six ways, as depicted in FIGS. 14A to 14F. In all cases, themagnitude of |i₁| is different from |i₀|. In order to remove the DCcurrent, the asymmetry between the positive and negative half cycles incurrent waveform must be eliminated. The GSC scheme ensures that thesampled currents at the beginning of even half cycles (i_(2m)) approach−i_(ref) and the sampled currents at the beginning of odd half cycles(i_(2m+1)) approach +i_(ref) for all six cases. Thus, the GSC schemeeffectively eliminates any asymmetry in the current waveform and rendersthe DC current zero. In FIGS. 14A to 14F, the sampled currents of evenhalf cycles can have three different states as follows:

$\begin{matrix}\left\{ \begin{matrix}{{i\left\lbrack {2m} \right\rbrack} = {i\left\lbrack {{2m} + 2} \right\rbrack}} & {{{if}\mspace{14mu}{\varphi\left\lbrack {{2m} + 1} \right\rbrack}} = {\varphi\left\lbrack {2m} \right\rbrack}} \\{{i\left\lbrack {2m} \right\rbrack} < {i\left\lbrack {{2m} + 2} \right\rbrack}} & {{{if}\mspace{14mu}{\varphi\left\lbrack {{2m} + 1} \right\rbrack}} > {\varphi\left\lbrack {2m} \right\rbrack}} \\{{{i\left\lbrack {2m} \right\rbrack} > {i\left\lbrack {{2m} + 2} \right\rbrack}}\ } & {{{if}\mspace{14mu}{\varphi\left\lbrack {{2m} + 1} \right\rbrack}} < {\varphi\left\lbrack {2m} \right\rbrack}}\end{matrix} \right. & (74)\end{matrix}$

From the GSC control law:

${{\varphi\left\lbrack {{2m} + 1} \right\rbrack} - {\varphi\left\lbrack {2m} \right\rbrack}} = {{0.5}\frac{L_{s}f_{s}}{kV_{b}}\Delta{{i\left\lbrack {2m} \right\rbrack}.}}$

Therefore, the states in Eq'n (74) can be rewritten as:

$\begin{matrix}\left\{ \begin{matrix}{{i\left\lbrack {2m} \right\rbrack} = {i\left\lbrack {{2m} + 2} \right\rbrack}} & {{{if}\mspace{14mu}{i\left\lbrack {2m} \right\rbrack}} = {- i_{ref}}} \\{{i\left\lbrack {2m} \right\rbrack} < {i\left\lbrack {{2m} + 2} \right\rbrack}} & {{{{if}\mspace{14mu}{i\left\lbrack {2m} \right\rbrack}} < {- i_{ref}}}\mspace{14mu}} \\{{{i\left\lbrack {2m} \right\rbrack} > {i\left\lbrack {{2m} + 2} \right\rbrack}}\ } & {{{if}\mspace{14mu}{i\left\lbrack {2m} \right\rbrack}} > {- i_{ref}}}\end{matrix} \right. & (75)\end{matrix}$

A similar procedure is used to find the three different states for thesampled currents of odd half cycles as follows:

$\begin{matrix}\left\{ \begin{matrix}{{i\left\lbrack {{2m} + 1} \right\rbrack} = {i\left\lbrack {{2m} + 3} \right\rbrack}} & {{{if}\mspace{14mu}{i\left\lbrack {{2m} + 1} \right\rbrack}} = i_{ref}} \\{{i\left\lbrack {{2m} + 1} \right\rbrack} < {i\left\lbrack {{2m} + 3} \right\rbrack}} & {{{{if}\mspace{14mu}{i\left\lbrack {{2\; m} + 1} \right\rbrack}} < i_{ref}}\mspace{14mu}} \\{{{i\left\lbrack {{2m} + 1} \right\rbrack} > {i\left\lbrack {{2m} + 3} \right\rbrack}}\ } & {{{if}\mspace{14mu}{i\left\lbrack {{2m} + 1} \right\rbrack}} > i_{ref}}\end{matrix} \right. & (76)\end{matrix}$

Table 3 shows that i[2m] approaches −i_(ref) and i[2m+1] approachesi_(ref) for all six cases in FIG. 14. Note that this is only the casewhen the sampled current is delayed by a half cycle (Δi[n]). If thesample current is not delayed; that is, if (Δi[n]) is not used in theGSC formulas, then i[n] will not converge to the reference currents forall six states. Moreover, some states will act as a feed-forward,causing i_(p) to diverge.

TABLE 3 case condition i[2m] condition i[2m + 1] FIG. 14A i[0] >−i_(ref) decreases i[1] < i_(ref) increases FIG. 14B i[0] > −i_(ref)decreases i[1] > i_(ref) decreases FIG. 14C i[0] < −i_(ref) increasesi[1] < i_(ref) increases FIG. 14D i[0] < −i_(ref) increases i[1] >i_(ref) decreases FIG. 14E i[0] < −i_(ref) increases i[1] < i_(ref)increases FIG. 14F i[0] > −i_(ref) decreases i[1] > i_(ref) decreases

Derivation of GSC for VDM

FIGS. 15A and 15B depict the deviation of i_(p) in the VDM scheme in oneswitching cycle (i.e., different D_(a)s for each half cycle) whenv_(a)>kv_(b). Since v_(a)>kv_(b), D_(a) is used as the control variable,the time intervals t₁ and t₂ must be varied by ΔD_(a) [n+1]/2 so that φand D_(b) remain constant, when D_(a)[n+1] changes by ΔD_(a)[n]. (Thisrelation can be inferred from (11), (12), (13)). As shown in FIGS. 15Aand 15B, the currents are sampled at i[n], i[n+1], and i[n+2], whichcorrespond to the instants when v_(p) changes from zero to +v_(a) or−v_(a). Additionally, i[n+2]=i[n] when D_(a)[n+1]=D_(a)[n].

FIG. 15A shows that, by increasing D_(a), [n+1], i[n+2] also increases.For this figure, the following relationship can be derived:

$\begin{matrix}{\frac{{i\lbrack n\rbrack} - {i\left\lbrack {n + 2} \right\rbrack}}{T_{s}} = {{\left( {m_{2} - m_{3}} \right)\Delta{D_{a}\lbrack n\rbrack}} + {\left( {m_{1} - m_{3}} \right){D\lbrack n\rbrack}\mspace{14mu}{where}}}} & (77) \\{{\Delta{D_{a}\lbrack n\rbrack}} = {{D_{a}\left\lbrack {n + 1} \right\rbrack} - {D_{a}\lbrack n\rbrack}}} & (78)\end{matrix}$

Then, by using (47), (29), (52), (30), and (77), the followingrelationship is derived:

$\begin{matrix}{\frac{{- d}{i\lbrack n\rbrack}}{dt} = {\frac{V_{a} + {kV_{b}}}{L_{s}}\Delta{D_{a}\lbrack n\rbrack}}} & (79)\end{matrix}$

In FIG. 15B, i[n+2] decreases as D_(a), [n+1] increases, and if D_(a),[n]=D_(a), [n+1], then i[n+2]=i[n]. The current derivative may bewritten as follows:

$\begin{matrix}{\frac{d{i\lbrack n\rbrack}}{dt} = {\frac{V_{a} + {kV_{b}}}{L_{s}}\Delta{D_{a}\lbrack n\rbrack}}} & (80)\end{matrix}$

Then, by using (56), (79), and (80) the following difference equationcan be achieved:

$\begin{matrix}{\frac{d\Delta{i\lbrack n\rbrack}}{dt} = {{- \frac{V_{a} + {kV_{b}}}{L_{s}}}\Delta{D_{a}\lbrack n\rbrack}}} & (81)\end{matrix}$

By using (60) and (81), the GSC control law for VDM when v_(a)>kv_(b) isderived as:

$\begin{matrix}{\frac{{0.5}\Delta{i\lbrack n\rbrack}}{dt} = {\frac{V_{a} + {kV_{b}}}{L_{s}}\Delta{D_{a}\lbrack n\rbrack}}} & (82) \\{{D_{a}\left\lbrack {n + 1} \right\rbrack} = {{\frac{{0.5}L_{s}f_{s}}{V_{a} + {kV_{b}}}\Delta{i\lbrack n\rbrack}} + {D_{a}\lbrack n\rbrack}}} & (83)\end{matrix}$

FIGS. 16A and 16B depict the deviation of i_(p) for the VDM scheme inone switching cycle (i.e., different D_(b)s for each half cycle) whenv_(a)<kv_(b). Again, the time intervals t₁ and t₂ must be varied byΔD_(a)[n+1]/2 so that φ and D_(a) remain constant when D_(b) [n+1]changes by ΔD_(b)[n]. The currents are sampled at i[n], i[n+1], andi[n+2], which correspond to the the instants when kv_(s) changes from+kv_(b) or −kv_(a) to zero. Additionally, when D_(a)[n+1]=D_(a)[n],i[n+2]=i[n].

FIG. 16A shows that, by decreasing D_(b)[n+1], i[n+2] also decreases,representing the following relationship:

$\begin{matrix}{\frac{{i\lbrack n\rbrack} - {i\left\lbrack {n + 2} \right\rbrack}}{T_{s}} = {{\left( {m_{4} - m_{2}} \right)\Delta{D_{a}\lbrack n\rbrack}} + {\left( {m_{1} + m_{2}} \right){D_{a}\lbrack n\rbrack}\mspace{14mu}{where}}}} & (84) \\{{\Delta{D_{b}\lbrack n\rbrack}} = {{D_{b}\left\lbrack {n + 1} \right\rbrack} - {D_{b}\lbrack n\rbrack}}} & (85)\end{matrix}$

By using (47), (29), (52), (31), and (84), the current derivative can becalculated as:

$\begin{matrix}{\frac{{- d}{i\lbrack n\rbrack}}{dt} = {\frac{V_{a} + {kV_{b}}}{L_{s}}\Delta{D_{b}\lbrack n\rbrack}}} & (86)\end{matrix}$

In FIG. 16B, i[n+2] increases as D_(a)[n+1] decreases, and ifD_(a)[n]=D_(a)[n+1], then i[n+2]=i[n]. The current derivative can bewritten as:

$\begin{matrix}{\frac{d{i\lbrack n\rbrack}}{dt} = {\frac{V_{a} + {kV_{b}}}{L_{s}}\Delta{D_{b}\lbrack n\rbrack}}} & (87)\end{matrix}$

Then, using (56), (86), and (87) the following difference equation isderived:

$\begin{matrix}{\frac{d\;\Delta\;{i\lbrack n\rbrack}}{dt} = {{- \frac{V_{a} + {kV}_{b}}{L_{s}}}\Delta\;{D_{b}\lbrack n\rbrack}}} & (88)\end{matrix}$

By using (60) and (88), the GSC control law for VDM when v_(a)<kv_(b) isgiven by:

$\begin{matrix}{\frac{0.5\Delta\;{i\lbrack n\rbrack}}{dt} = {\frac{V_{a}{kV}_{b}}{L_{s}}\Delta\;{D_{b}\lbrack n\rbrack}}} & (89) \\{{D_{b}\left\lbrack {n + 1} \right\rbrack} = {{\frac{0.5L_{s}f_{s}}{V_{a} + {kV}_{b}}\Delta\;{i\lbrack n\rbrack}} + {D_{b}\lbrack n\rbrack}}} & (90)\end{matrix}$

Derivation of GSC for TCM

FIGS. 17A and 17B depict the deviation of i_(p) for the TCM scheme inone switching cycle (i.e., different φ[n]s for each half cycle) whenv_(a)>kv_(b). The currents are sampled at i[n], i[n+1], and i[n+2],which correspond to the instants when v_(p) changes from +v_(a) or−v_(a) to zero. Again, i[n+2]=i[n] when D_(a), [n+1]=D_(a)[n]. FIG. 17Ashows that by increasing φ[n+1], i[n+2] decreases and the followingrelationship can be derived:

$\begin{matrix}{{\frac{{i\lbrack n\rbrack} - {i\left\lbrack {n + 2} \right\rbrack}}{T_{s}} = {{{{- m_{s}}\Delta\;{\varphi_{\Delta}\lbrack n\rbrack}} + {m_{2}\Delta\;{D_{a}\lbrack n\rbrack}}} = {{- 2}m_{3}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}}}{where}} & (91) \\{{{\Delta\varphi}_{\Delta}\lbrack n\rbrack} = {{\varphi\left\lbrack {n + 1} \right\rbrack} - {\varphi\lbrack n\rbrack}}} & (92)\end{matrix}$

By using (52), (30), (91), and (92), the current derivative iscalculated as:

$\begin{matrix}{\frac{- {{di}\lbrack n\rbrack}}{dt} = {\frac{2{kV}_{b}}{L_{s}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (93)\end{matrix}$

In FIG. 17B, i[n+2] increases as φ[n+1] increases, and if φ[n]=φ[n+1],then i[n+2]=i[n]. The current derivative can be written as:

$\begin{matrix}{\frac{{di}\lbrack n\rbrack}{dt} = {\frac{2{kV}_{b}}{L_{s}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (94)\end{matrix}$

By using (56), (91), and (94), the following difference equation can bedetermined:

$\begin{matrix}{\frac{d\;\Delta\;{i\lbrack n\rbrack}}{dt} = {{- \frac{2{kV}_{b}}{L_{s}}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (95)\end{matrix}$

By using (60) and (95), the GSC control law for TCM when v_(a)>kv_(b)can be derived as follows:

$\begin{matrix}{\frac{0.5\Delta\;{i\lbrack n\rbrack}}{dt} = {\frac{2{kV}_{b}}{L_{s}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (96) \\{{\varphi\left\lbrack {n + 1} \right\rbrack} = {{\frac{0.5L_{s}f_{s}}{2{kV}_{b}}\Delta\;{i\lbrack n\rbrack}} + {\varphi\lbrack n\rbrack}}} & (97)\end{matrix}$

FIGS. 18A and 18B depict the deviation of i_(p) for the TCM scheme inone switching cycle (i.e., different φ[n]s for each half cycle) whenv_(a)<kv_(b). The currents are sampled at i[n], i[n+1], and i[n+2],which correspond to the instants when kv_(s) changes from zero to+kv_(b) or −kv_(b). Again, i[n+2]=i[n] when D_(a)[n+1]=D_(a)[n]. FIG.18A shows that by increasing φ[n+1], i[n+2] decreases. The followingrelationship can be derived:

$\begin{matrix}{\frac{{i\lbrack n\rbrack} - {i\left\lbrack {n + 2} \right\rbrack}}{T_{s}} = {{{m_{4}\Delta\;{\varphi_{\Delta}\lbrack n\rbrack}} + {m_{2}\Delta\;{D_{b}\lbrack n\rbrack}}} = {2m_{4}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}}} & (98)\end{matrix}$

By using (52), (30), (98), and (92), the current derivative iscalculated as:

$\begin{matrix}{\frac{- {{di}\lbrack n\rbrack}}{dt} = {\frac{2V_{a}}{L_{s}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (99)\end{matrix}$

In FIG. 18B, i[n+2] increases as φ[n+1] increases and if φ[n]=φ[n+1],then i[n+2]=i[n]. The current derivative can then be written as:

$\begin{matrix}{\frac{{di}\lbrack n\rbrack}{dt} = {\frac{2V_{a}}{L_{s}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (100)\end{matrix}$

By using (56), (99), and (100), the following difference equation isgiven:

$\begin{matrix}{\frac{d\;\Delta\;{i\lbrack n\rbrack}}{dt} = {{- \frac{{- 2}V_{a}}{L_{s}}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (101)\end{matrix}$

By using (60) and (101), the GSC control law for TCM when v_(a)<kv_(b)can be derived as:

$\begin{matrix}{\frac{0.5\Delta\;{i\lbrack n\rbrack}}{dt} = {\frac{2V_{a}}{L_{s}}{{\Delta\varphi}_{\Delta}\lbrack n\rbrack}}} & (102) \\{{\varphi\left\lbrack {n + 1} \right\rbrack} = {{\frac{0.5L_{s}f_{s}}{2V_{a}}\Delta\;{i\lbrack n\rbrack}} + {\varphi\lbrack n\rbrack}}} & (103)\end{matrix}$

Stability and Robustness of the GSC Control System

The well-known Lyapunov function can be used to investigate stabilityand robustness of differential equations. In the current case, theLyapunov function is defined as:

V=Δi _(p,t) ₀ ²[k]  (104)

Additionally, note that the control laws given above can be derivedbased on Eq'n (61). Therefore, the following relationship in the currenterror holds when GSC control is used:

Δi[n+1]=0.5Δi[n]  (105)

The difference of the Lyapunov function can then be derived as follows:

$\begin{matrix}\begin{matrix}{{{V\lbrack n\rbrack} - {V\left\lbrack {n - 1} \right\rbrack}} = {{\Delta\;{i^{2}\lbrack n\rbrack}} - {\Delta\;{i^{2}\left\lbrack {n - 1} \right\rbrack}}}} \\{= {\left( {0.5\Delta\;{i\lbrack n\rbrack}} \right)^{2} - {\Delta\;{i^{2}\left\lbrack {n - 1} \right\rbrack}}}} \\{= {{- 0.75}\Delta\;{i^{2}\lbrack n\rbrack}}}\end{matrix} & \lbrack 106\rbrack\end{matrix}$

which shows that V[n]−V[n−1] is always negative. Therefore, Δi isasymptotically stable at origin, meaning Δi converges to zero atsteady-state.

In practice, there may be uncertainties in the system. As an example,the value of external series inductance might vary during differentoperations. Also, the voltage measurement of the DC bus might have somedeviation from its real value. A robust control system preferablyprovides stability even in light of such uncertainties and fluctuations.The GSC control laws of the present invention depend on the transformerratio, output voltage, switching period, and the series inductance. Bytaking into account the percentage error of the series inductance(ΔL_(s)%), the relation between the two consecutive errors can bederived as follows:

Δi _(p)[k+1]=(0.5−0.5θL _(s)%)Δi _(p)[k]  (107)

Rewriting the difference equation for the Lyapuov function thus yields:

V[k]−V[k−1]=¼(ΔL _(s)%²−2ΔL _(s)%−3)Δi _(p,t) ₀ ²[k]  (108)

The above equation is negative if

−100%<ΔL _(s)%<300%→−L _(s) <ΔL _(s)<3L _(s)  (109)

Therefore, if the perturbations of L_(s) are within the range in Eq'n(109), the control system is still asymptotically stable. Similarresults can be achieved for variations in switching frequency.

The perturbation in sensing the DC bus voltage can be investigated bytaking into account the percentage error (ΔV %) when sensing the DC busvoltage. The relationship between the two consecutive errors can bederived as follows:

$\begin{matrix}{{\Delta\;{i_{p}\left\lbrack {k + 1} \right\rbrack}} = {\left( {1 - \frac{0.5}{1 + {\Delta\; v_{b}\%}}} \right)^{2}\Delta\;{i_{p}\lbrack k\rbrack}}} & (110)\end{matrix}$

leading to the following Lyapunov function:

$\begin{matrix}{{{V\lbrack k\rbrack} - {V\left\lbrack {k - 1} \right\rbrack}} = {\left( {\left( {1 - \frac{0.5}{1 + {\Delta\; v_{b}\%}}} \right)^{2} - 1} \right)\Delta\;{i_{p,t_{0}}^{2}\lbrack k\rbrack}}} & (111)\end{matrix}$

The above equation is negative if

−75%<Δv _(b)→  (112)

−0.75v _(b) <Δv _(b)  (113)

Therefore, if the perturbations of v_(b) are compliant with the range in(113), then the control system is asymptotically stable. Thus, the GSCcontrol system of the present invention illustrates high robustnessagainst system uncertainties and fluctuations in the system.

The stability and robustness of the GSC control law for a general casewith different scale factors (λ) can be written

Δi[n]=(1−λ)Δi[n−1]  (114)

Following a similar procedure as that above for the scale factor ofλ=0.5, it can be derived that when the scale factor is within 0<λ<2, theerror reduces in a geometric sequence progression towards zero withcommon ratio of 1−λ.

Simulation and Example

The performance of the control system according to one aspect of thepresent invention was verified by simulation results, using theprototype of FIG. 2. Again, as should be clear, nothing in this examplesimulation should be taken as limiting the invention in any way.

PSIM™ simulation software was used to conduct simulations for a DABDC/DC converter with specifications given in Table 4.

TABLE 4 Symbol Parameter Value V_(a) Input Voltage 100 V_(DC) V_(b)Output Voltage 150 V_(DC) f_(s) Switching Frequency 100 kHz L_(s) SeriesInductor 10 μH C_(A) Input Capacitance 160 μF C_(B) Output Capacitor 160μF n Transformer's Turns 0.9 Ratio S MOSFETs SQM10250E_GE3

As an example, the control system was implemented inside a Cyclone™ IVFPGA (field-programmable gate array). VHSIC Hardware DescriptionLanguage was used to program the FPGA. A phase-locked loop block (“PLLblock) inside the FPGA was used to boost the clock of the externaloscillator from 24 MHz to 120 MHz. The block diagram of FIG. 2 shows apower flow from port A to port B. The analogue-to-digital (ADC) devicesused were ADS7884™ devices. The current transformer used was PN: P0581N,optimized for frequencies of 50-500 kHz. The corresponding ADC fori_(p)[n] performs sampling every half cycle. This timing is controlledby a sampling signal (ss) which is generated inside the modulator basedon the values in Table 4. ADS7884™ devices are capable of sampling 3MSPS, and thus can support the required half cycle (200 kSPS) sensingfor the i_(p) value. The sampled currents were kept for every threeconsecutive half cycles of (i[n]). Based on this calculated power level,the modulator also operates in three different modes: TCM, VDM, and PSM.These modulation schemes were constructed using a Mealy state machine.

It should, however, be clear that the digital control system asexplained above may be implemented using various platforms such as FPGA,digital signal processors (“DSP”), and others.

FIGS. 19A to 19D depicts the transient response of the GSC controlsystem. Specifically, FIG. 19A depicts the current waveforms i_(p) andI_(b) for a positive and negative step change in the inner loopreference current. At t=3 ms, the reference current is changed such thati_(b) changes from 9 A to 6 A. Then, at t=5 ms, the reference current ischanged such that i_(b) changes from 6 A to 10 A. In both cases, i_(p)changes symmetrically in response to step change in reference current.

FIG. 19B depicts the transition of current at t=5 ms in enlarged scale.According to this figure, i_(p) shows a symmetrical growth. FIG. 18Cdepicts the current error during this time interval. As can be seen,current error reduces by half in every switching cycle (as seen in Eq'n(60)). FIG. 19D depicts the smooth transition of the control variable(φ) in its trend towards steady state value of 0.19.

FIGS. 20A to 20C depict the control system performance when the seriesinductor is varied by 100% (L_(p)=20 μH). In this scenario, GSC shows aslower transient response compared to L=10 μH. Further, as can be seen,the GSC control system reduces the current error to zero, which showsthe high robustness of the proposed control against uncertainties andfluctuations of the inductor.

FIG. 21 is a plot showing the transient response of the GSC controlsystem of the present invention, as compared to conventional PIcontrollers for DAB converters. As can be seen, the present inventionprovides faster response than conventional schemes.

FIG. 22 illustrates the transient operation of the converter inconventional application, where the control variables are changed everyswitching cycle (i.e., symmetric half cycle modulation scheme isemployed). As can be seen, in this scenario, i_(p) changesasymmetrically and the peak current can potentially rise to high values,causing temporary saturation of the transformer, as discussed above.

FIG. 23 depicts the steady-state operation of the converter when poweris transferred from port B to port A. It can be seen that in thisscenario, φ is negative. The steady state equation of power flow is thesame as in power flow from port A to port B given in (10), but withnegative sign. That is, reversing the direction of power flow in thesystem of FIG. 2 only changes the sign term, while the rest of therelations derived above remain the same.

Referring now to FIG. 24, a flowchart illustrating a method according toone aspect of the invention is illustrated. The transformer primarycurrent is measured every half switching cycle and modified by thefeedback unit (step 2400). The inner loop reference signal is subtractedfrom the modified feedback signal to create the inner loop error (step2410). The error is used by the GSC compensator (the inner loopcompensator) to create the control parameters for the modulator (step2420). Then, based on those control parameters, a modulation techniqueis applied to change the switching signals (step 2430). The modulationtechnique is one of PSM, VDM, or TCM, as detailed above. The modifiedswitching signals changes the transformer primary current until itreaches the desired inner loop reference current (inner loop errorbecomes zero). The resulting feedback loop reduces error and increasesreliability and efficiency of the DAB converter over a broad range ofinput conditions.

As noted above, for a better understanding of the present invention, thefollowing references may be consulted:

-   [1] Krismer and Kolar, “Closed Form Solution for Minimum Conduction    Loss Modulation of DAB Converters,” IEEE Transactions on Power    Electronics, vol. 27, DOI 10.1109/TPEL.21011.2157976, no. 1, pp.    174-188, January 2012.-   [2] F. Krismer and J. W. Kolar, “Accurate Small-Signal Model for the    Digital Control of an Automotive Bidirectional Dual Active Bridge,”    IEEE Transactions on Power Electronics, vol. 24, DOI    10.1109/TPEL.2009.2027904, no. 12, pp. 2756-2768, December 2009.-   [3] F. Krismer and J. W. Kolar, “Accurate Power Loss Model    Derivation of a High-Current Dual Active Bridge Converter for an    Automotive Application,” IEEE Transactions on Industrial    Electronics, vol. 57, DOI 10.1109/TIE.2009.2025284, no. 3, pp.    881-891, March 2010.-   [4] F. Krismer, S. Round, and J. W. Kolar, “Performance optimization    of a high current dual active bridge with a wide operating voltage    range,” in 2006 37th IEEE Power Electronics Specialists Conference,    DOI 10.1109/pesc.2006.1712096, pp. 1-7, June 2006.

It should be clear that the various aspects of the present invention maybe implemented as software modules in an overall software system. Assuch, the present invention may thus take the form of computerexecutable instructions that, when executed, implement various softwaremodules with predefined functions.

Further, as used herein, the expression “at least one of X and Y” meansand should be construed as meaning “X or Y or both X and Y”.

A person understanding this invention may now conceive of alternativestructures and embodiments or variations of the above all of which areintended to fall within the scope of the invention as defined in theclaims that follow.

We claim:
 1. A control system for a dual active bridge (DAB) DC/DCconverter, said control system comprising: a feedback unit; acompensator; and a modulator, wherein said feedback unit measures atransformer current from a bridge component of said DAB converter asinput and wherein an inner loop reference current is subtracted from anoutput of said feedback unit; wherein an output of said subtraction ispassed to said compensator; wherein said compensator determines controlparameters for said modulator; and wherein said modulator applies amodulation technique based on said control parameters to change aswitching pattern to thereby control said transformer current and tothereby control a power flow in said circuit.
 2. The control systemaccording to claim 1, wherein said modulation technique is one of:phase-shift modulation (PSM), variable duty cycle modulation (VDM), andtriangular current mode modulation (TCM).
 3. The control systemaccording to claim 1, wherein said feedback unit samples saidtransformer current once every half switching cycle.
 4. The controlsystem according to claim 1, wherein said compensator updates saidcontrol parameters once every half switching cycle.
 5. The controlsystem according to claim 1, wherein said modulator updates saidswitching pattern every half switching cycle.
 6. The control systemaccording to claim 1, wherein said compensator determines said controlparameters using a geometric-sequence-control (GSC) method.
 7. Thecontrol system according to claim 6, wherein said GSC method is based onan error in said inner loop reference current.
 8. The control systemaccording to claim 6, wherein said GSC method is applied such that anerror in said inner loop reference current decreases in a geometricsequence progression over successive half switching cycle samplings, andwherein said control parameters are by-products of said GSC method. 9.The control system according to claim 1, wherein said control parameterscomprise at least one of: a phase shift of the converter, a duty cycleof said bridge component, and a duty cycle of another bridge componentof said DAB converter.
 10. The control system according to claim 1,wherein said system conforms to at least one of the followingconditions: when a power level within said circuit is below a firstthreshold, said modulation technique is triangular current modemodulation (TCM); when said power level is between said first thresholdand a second threshold, said modulation technique is variable duty cyclemodulation (VDM), said second threshold being higher than said firstthreshold; and when said power level is above said second threshold,said modulation technique is phase-shift modulation (PSM).
 11. A methodfor controlling a power flow within a circuit comprising a dual activebridge (DAB) DC/DC converter, said method comprising the steps of:receiving a current from a bridge component of said DAB converter asinput for a compensator; measuring a current from a transformercomponent of said DAB converter; based on said current, determiningcontrol parameters for a modulator; and using said modulator, applying amodulation technique based on said control parameters, to thereby modifysaid current and to thereby control said power flow in said circuit. 12.The method according to claim 11, wherein said modulation technique isone of: phase-shift modulation (PSM), variable duty cycle modulation(VDM), and triangular current mode modulation (TCM).
 13. The methodaccording to claim 11, wherein said control parameters are determined byapplying a geometric-sequence-control (GSC) method.
 14. The methodaccording to claim 13, wherein said GSC method is based on inner loopcurrent error.
 15. The method according to claim 11, wherein saidcontrol parameters are determined by a compensator.
 16. The methodaccording to claim 15, wherein said compensator updates said controlparameters once every half switching cycle.
 17. The method according toclaim 13, wherein said GSC method is applied such that an inner loopcurrent error decreases in a geometric sequence progression oversuccessive half switching cycle samplings, and wherein said controlparameters are by-products of said GSC method.
 18. The method accordingto claim 11, wherein said control parameters comprise at least one of: aphase shift of the converter, a duty cycle of said bridge component, anda duty cycle of another bridge component of said DAB converter.
 19. Themethod according to claim 11, wherein said method conforms to at leastone of the following conditions: when a power level within said circuitis below a first threshold, said modulation technique is triangularcurrent mode modulation (TCM); when said power level is between saidfirst threshold and a second threshold, said modulation technique isvariable duty cycle modulation (VDM), said second threshold being higherthan said first threshold; or when said power level is above said secondthreshold, said modulation technique is phase-shift modulation (PSM).20. Non-transitory computer-readable media having encoded thereoncomputer-readable and computer-executable instructions that, whenexecuted, implement a method for controlling a power flow within acircuit comprising a dual active bridge (DAB) DC/DC converter, saidmethod comprising the steps of: receiving a current from a bridgecomponent of said DAB converter as input for a compensator; measuring acurrent from a transformer component of said DAB converter; based onsaid current, determining control parameters for a modulator; and usingsaid modulator, applying a modulation technique based on said controlparameters, to thereby modify said current and to thereby control saidpower flow in said circuit.